Let's solve the given system of equations:
[tex]\[
\left\{
\begin{array}{l}
\frac{x}{2} = \frac{y}{3} = \frac{2}{4} \\
2x - 3y - 4z = -42
\end{array}
\right.
\][/tex]
First, we can simplify the fraction on the right-hand side.
[tex]\[
\frac{2}{4} = \frac{1}{2}
\][/tex]
So, we have:
[tex]\[
\frac{x}{2} = \frac{y}{3} = \frac{1}{2}
\][/tex]
From this, we can derive:
[tex]\[
\frac{x}{2} = \frac{1}{2}
\][/tex]
Multiplying both sides by 2:
[tex]\[
x = 1
\][/tex]
Similarly, from:
[tex]\[
\frac{y}{3} = \frac{1}{2}
\][/tex]
Multiplying both sides by 3:
[tex]\[
y = \frac{3}{2} = 1.5
\][/tex]
Now, we substitute [tex]\(x = 1\)[/tex] and [tex]\(y = 1.5\)[/tex] into the second equation:
[tex]\[
2x - 3y - 4z = -42
\][/tex]
Substituting [tex]\(x = 1\)[/tex] and [tex]\(y = 1.5\)[/tex]:
[tex]\[
2(1) - 3(1.5) - 4z = -42
\][/tex]
Simplifying within the equation:
[tex]\[
2 - 4.5 - 4z = -42
\][/tex]
[tex]\[
-2.5 - 4z = -42
\][/tex]
Next, isolate [tex]\(z\)[/tex] by first adding 2.5 to both sides:
[tex]\[
-4z = -42 + 2.5
\][/tex]
[tex]\[
-4z = -39.5
\][/tex]
Now, divide both sides by -4:
[tex]\[
z = \frac{-39.5}{-4}
\][/tex]
[tex]\[
z = 9.875
\][/tex]
So, the solutions to the system of equations are:
[tex]\[
x = 1.0000,\quad y = 1.5000,\quad z = 9.8750
\][/tex]
